Orbit counting with an isometric direction. (English) Zbl 1115.37015

Kolyada, S. (ed.) et al., Algebraic and topological dynamics. Proceedings of the conference, Bonn, Germany, May 1–July 31, 2004. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3751-6/pbk). Contemp. Math. 385, 293-302 (2005).
Let \(\phi: x\mapsto 2x\) be the map on the ring \(Y= \mathbb Z[1/3]\). Let \(X\) be the dual (character) group for \(Y\), \(f\) be the dual map. The pair \((X,f)\) is studied in the paper; the map \(f\) is the 3-adic extension of the circle doubling map. It has a 3-adic eigendirection in which it behaves like an isometry. The loss of hyperbolicity leads to some asymptotic result on orbit counting, weaker than those obtained for dynamical systems with hyperbolic behavior and which are analogous to the prime number theorem and Mertens’ theorem.
For the entire collection see [Zbl 1075.37001].


37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37E10 Dynamical systems involving maps of the circle
11N05 Distribution of primes
37P35 Arithmetic properties of periodic points
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